The fire, which started about 8pm (0100 GMT) on Friday from a fuel pipe at Pertamina's Plumpang fuel storage depot in the capital Jakarta, quickly spread to nearby houses and sent residents in the densely populated area into panic.
Authorities initially put the death toll at 17 but revised it later to 13.
Dozens were injured and hundreds were evacuated.
The fire had been extinguished by early morning hours on Saturday, North Jakarta firefighter official Abdul Wahid said.
"I have ordered Pertamina to immediately investigate this case thoroughly," State-Owned Enterprise Minister Erick Thohir said via Instagram.
"There must be an operational review."
Sugeng Suparwoto, who heads the parliament's energy committee, called for an audit of Pertamina's facilities.
"All facilities, whether refineries or storage, must be audited again," he said on KompasTV, noting that Pertamina often had fire incidents at its facilities.
In 2021, a major fire broke out at Pertamina's refineries in Balongan and Cilacap.
Sugeng said there should be a bigger distance between Pertamina's storage facilities and residential areas.
"For a facility with Plumpang's capacity, there should be at least one to two kilometres distance with residential area."
Plumpang, with a storage capacity of more than 300,000 kilolitres, is one of Pertamina's biggest fuel terminals.
Dense residential area stands outside the Plumpang's outer wall, separated only by a narrow street, a Reuters witness said.
Local residents could smell the fuel about 30 minutes before the fire, Abdul Syukur, who lives nearby, told KompasTV.
"The smell was so strong there were people throwing up and some nearly fainted," he said.
CEO Nicke Widyawati said Pertamina would launch an internal review while ensuring Jakarta's fuel supply would remain secure as the company diverted supplies from other terminals.
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